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Unformatted text preview: Physics 4D Lab Exercises page 1 Physics 4D Lab: Electron Spin Resonance The objective of this experiment is to determine the g factor of the electron using the experimental technique called electron spin resonance. Introduction: A . Review of Classical Ideas: (Note: when vector notation is desireable vectors are in bold) According to classical physics, the magnetic dipole moment μ of a circular current loop with current i and cross-section area A is defined by μ = i A with a direction normal to the area in the same sense as that indicated by the right hand rule for conventional current. We can apply this definition to the planetary model of an electron of mass m , charge q = -e , orbit radius r , orbit period T and velocity v orbiting about a positively charged nucleus. The current associated with the orbit will be q/T and the velocity will be v = 2 π r/T. The magnitude of the angular momentum L of the electron in its orbit is l = mvr as shown in the diagram to the right The magnitude of the magnetic moment μ of the electron of charge q = -e would then be: μ = i A = (q/T)( π r 2 ) = q / (2 π r/v) ( π r 2 ) = (q/2m)(m v r) or μ = (q /2m) L (Eq.1) or in terms of vectors: μ = (-e /2m) L (Eq.2) The negative charge would mean the direction of electron magnetic moment μ is opposite to its angular momentum L as depicted in the diagram. In general, the classical orbit of an electron will be directed in an arbitrary direction in space. The effect of an external magnetic field B on an arbitrarily oriented orbit will be to exert a torque τ on the system given by τ = μ B sin θ with a direction perpendicular to the plane of μ and B vector as shown in the diagram to the right. Since the direction of the torque is perpendicular to the angular momentum vector, there will be no change in the magnitude of the angular momentum and but only be changes in the direction of the angular momentum. As we know from classical mechanics torques produce infinitesmal changes in angular momentum dL of a rotating system according to : | d L |= | τ |dt = | μ x B |dt = μ B sin θ d t . Referring to the diagram to the right, the result is a precession of the angular momentum vector L about an axis defined by the direction of the magnetic field with an angular velocity ω L = d α /dt where d α is given by d α = dL/L sin θ = ( μ B sin θ d t )/L sin θ = ( μ B /L) dt. The precessional angular velocity would then be given by: ω L = d α /dt = μ B /L. (Eq.3) This example is analogous to the precession of a gyroscope in Physics 4D Lab Exercises page 2 a gravitational field. Substitution of μ from (Eq.1) into (Eq.3) gives ω L = e B /2 m (Eq.4) The frequency ω L (in radian/sec here) is called the Larmor precession frequency ω L . It is independent of θ and L The magnetic potential energy U m of a magnetic dipole in a magnetic field is given by : U mag = - μ • B =- μ B cos θ = μ z B (Eq.5) where θ = π /2 is chosen as the zero of energy. This suggests/2 is chosen as the zero of energy....
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